So, obviously there's a dearth of secondary literature on this, but one thing I can think of right off the bat is this interview with Koji Kondo. Kondo is arguably the most enduring composer of the class of people you're asking after. Is there a more iconic piece of early video game music than the first level of Super Mario Brothers? If so, it's the Legend of Zelda theme, which he also wrote.

Kondo speaks to your specific question a bit here, describing how he found his way to Nintendo after graduating from an art college, the Osaka School of the Arts:

That was actually the first year Nintendo started hiring for dedicated sound staff. My friend saw the ad first, and he told me about it....Back then, there was no MIDI or other protocol that could directly convert your keyboard playing to data. Everything was composed directly on the computer, entering notes manually. I did practice on my Electone at home and sketch out ideas, which I’d then take to the office and enter into the computer. I would do the arrangement on the computer there, adjusting things as I went.

Kondo remained the head composer for most of Nintendo's major releases from the 80s through to 1998's Ocarina of Time, after which he ascended to heaven, having achieved perfection moved on to an executive role in the company, though he still personally contributes to major titles.

By contrast, Nobou Uematsu, of the Final Fantasy series, has no formal musical training. In the linked interview, he describes his start at Squaresoft:

At the time, I was 24 or 25, and I wanted to work in music, but the world wasn’t forgiving enough to allow someone with no connections, ability, or experience to get a job like that. I knew I wanted to do something, so I made demo tapes every day and sent them to record companies, but I couldn’t get near a job. My home was just a small room, but every night people with different dreams would get together there and drink. People who wanted to be painters, writers, calligraphers, and such. That kind of thing was happening every night at my place. And one day there was a girl who said she was making games. Before long, she asked me if I could make tracks to be used in games, and the place she ended up taking me to was Square. Back then, Square was based really close to where I was living, so I ended up visiting there a lot. That was when I was about 25. Game music didn’t exist as a job back then as far as most people knew, so I thought I’d just do game music for a couple of years and then move onto something more proper, though maybe that isn’t the best word. Something like film music, or writing popular music. I thought about getting a job like that. So, I intended to join Square as a temporary job, but I’ve ended up staying at the company for about 25 years.

Either way, it looks like game companies weren't concerned with hiring established composers for their games, hiring new and possibly untrained talent.

/u/TRB1783 already addressed Japanese video game music in the early days, so I want to put in one or two words about Western video game music.

At Atari, many of the composers were considered programmers first, and many like David Whittaker and Tim Follin had essentially no formal training in music theory or composition. They were hired as programmers, and they just happened to also have a talent or interest in music as a hobby, so they were assigned to write the music for the games. As has been mentioned in the other answer, there was no MIDI or proper music encoding software in those days, so writing music for the Amiga or the Commodor64 required a fairly in-depth familiarity with the machine code and how it transformed impedance into sound. Whittaker has admitted in interviews that in the early 1980s his composing methods boiled down to:

Just plonking away on a keyboard and using what sounded half-decent.

But there were also some people like Rob Hubbard and Chris Granner who had formal musical training and became involved in the industry because of social connections to the burgeoning computer hobbyist communities. Hubbard did study music in college, although I'm not sure if he graduated. Granner was formally-trained organist with a BFA and everything. But my perception is that they were if not the exception, at least hardly the rule. On the whole, many of the early Western video game musicians, especially at Atari, were programmers first and musicians second. I have a perception, based on vague recollections of magazine articles I read as an adolescent, that sometime in the early 1990s, North American video game developers started hiring professional musicians. I'm having some trouble finding good sources for this, but I remember being impressed with the music for Civilization II when it came out, and Wiki says that Jeff Briggs, who has a PhD in Music composition, wrote most of the music for it.

Nearly all of the major players in the early video game music scene are still alive, and it now occurs to me that someone should write an oral history or something.

So in summary: North American and UK video games studios in the 1970s and 1980s largely (but not entirely!) were programmers by profession with some experience with music as a hobby. Sometime in the very late 1980s or early 1990s this started to change, and by the mid-1990s nearly all video game music is being written by people with formal training in music theory and composition.

So Let's say its the 80s, I made a video game, and now I want to hire someone to bop me out some chip tunes... Do I even hire someone, depending on budget? What am I looking for?

Many of the folks who were inputing music in these early-80s games were definitely not composers and did not have a music background; a lot of the time, the job would simply be done by the programmer himself. This is also why chiptune "arrangements" of licensing-free music (either classical or folk/traditional tunes) were surprisingly common! Of course, even in the 1980s, things did gradually develop in the direction of having a specialized figure that just dealt with the music. The book Game Sound by Karen Collins gives a greatly expanded account about these developments, including useful references.

Having recently spoken to one of the first people to ever put music in a video game, I wanted to answer this in a more technical way.

Until you got to the era that people are speaking about below, the mid-80s, almost all of the music done in games were by programmers, not even remotely trained people. They would sometimes have some decent tunes, but when you have monophonic or at most three channel sound there's not much use in hiring someone particularly good in that field. It's astounding to even hear something being produced electronically.

How sound was produced initially was by oscillating speakers at desired times per second to produce noises. It is literally the same way we do it today, but we have ways of distributing the sound to create volume controls and such. Most of the time they would do this in assembly or hex code, creating a routine which called on the speaker to be accessed when something happened.

When you get to the NES, you get both more professional composers and the ability to write music using independent programs, usually called 'trackers'. While that type of software wasn't as sophisticated as something like Famitracker that we use for chiptunes today, it's mostly the same idea. You're slightly more restricted in how much you can change the music to be truly unique, but there's enough versatility with a small number of sounds that it doesn't matter.

Electronic music was so new, they were calling on whatever they could think of, and they did amazing things with such limited channels. I particularly think Namco's composers like Junko Ozawa defined that age and deserve to be more well known.